Exponential decay for the semilinear wave equation with localized Kelvin-Voight damping

TL;DR

This paper proves exponential decay and stability for a semilinear viscoelastic wave equation with localized Kelvin-Voigt damping, using observability and unique continuation methods.

Contribution

It introduces a novel approach combining observability and unique continuation to establish exponential decay for the semilinear wave equation with localized damping.

Findings

Solutions decay exponentially in the weak phase space.

Global existence of solutions is established.

The method applies to locally damped viscoelastic wave equations.

Abstract

In the present paper, we are concerned with the semilinear viscoelastic wave equation subject to a locally distributed dissipative effect of Kelvin-Voigt type, posed on a bounded domain with smooth boundary. We begin with an auxiliary problem and we show that its solution decays exponentially in the weak phase space. The method of proof combines an observability inequality and unique continuation properties. Then, passing to the limit, we recover the original model and prove its global existence as well as the exponential stability.